New York University - ECON UA

Urban Economics

Problem Set #3 due Tuesday, Apr 5, in class

1. The third largest city of a country has a population of 2.5 million. Using the ranksize rule, what is the population of the 10^th largest city? 

Answer: The rank-size rule states that N_i=C/R_i, where N=population and R=rank of the ith city. C is a constant, i.e., the population of the largest city.

If the third largest city has a population of 2.5m, the largest one has 7.5m. Thus, the 10^th largest city has a population of 0.75m. 

2. Assume an urban cluster’s income (I) is given by the function I=4N, where N denotes the cluster’s population in million. The cluster’s diseconomies of aggregation (C) are given by C=0.5N². 
   1. Calculate the city’s utility maximizing population. 
   2. At this population, how large is the resulting utility? 
   3. Calculate the resulting utility if the population were on million higher and one million lower than the optimum. 

Answer: 

1. 1. if I=4N        à MI=4

      if C=0.5^N2   à MC=N

                              MI=MC  à N=4

1. 2. at N=4 à I=16 and C=8 à U=I-C=8 (c)            at N=5 à I=20 and C=12.5 à U=7.5

            at N=3 à I=12 and C=4.5 à U=7.5

3. Assume the utility functions for two cities are identical and are given by  U=N - 0.1*N², where N denotes the city’s population in million. 
   1. What is each city’s utility maximizing population?
   2. If each city had a population of 6.5 million people, how would these cities change their size? Assuming that the total population of 13 million cannot be changed, would there be a smaller and a larger city? Would there be three or more cities? Or would there be no change at all. Explain.  

Answers: 

1. 1. set MU equal to zero and solve for N

MU=1-0.2N=0   à 0.2N=1 à N=5

1. 2. A population of 6.5m is well above the utility maximizing population for each city. Nevertheless, the outcome is a stable equilibrium and there won’t be any change. The reasoning is the following. The first person that moves away from